[Barmpalias, George] Chinese Acad Sci, Inst Software, State Key Lab Comp Sci, Beijing 100190, Peoples R China. [Day, Adam R.] Univ Calif Berkeley, Dept Math, Berkeley, CA 94720 USA. [Lewis-Pye, Andy E. M.] Univ London London Sch Econ & Polit Sci, Dept Math, London WC2A 2AE, England.

Abstract:

The Turing degree of a real measures the computational difficulty of producing its binary expansion. Since Turing degrees are tailsets, it follows from Kolmogorov's 0-1 law that, for any property which may or may not be satisfied by any given Turing degree, the satisfying class will either be of Lebesgue measure 0 or 1, so long as it is measurable. So either the typical degree satisfies the property, or else the typical degree satisfies its negation. Further, there is then some level of randomness sufficient to ensure typicality in this regard. We describe and prove a large number of results in a new programme of research which aims to establish the (order theoretically) definable properties of the typical Turing degree, and the level of randomness required in order to guarantee typicality. A similar analysis can be made in terms of Baire category, where a standard form of genericity now plays the role that randomness plays in the context of measure. This case has been fairly extensively examined in the previous literature. We analyse how our new results for the measure-theoretic case contrast with existing results for Baire category, and also provide some new results for the category-theoretic analysis.

English Abstract:

The Turing degree of a real measures the computational difficulty of producing its binary expansion. Since Turing degrees are tailsets, it follows from Kolmogorov's 0-1 law that, for any property which may or may not be satisfied by any given Turing degree, the satisfying class will either be of Lebesgue measure 0 or 1, so long as it is measurable. So either the typical degree satisfies the property, or else the typical degree satisfies its negation. Further, there is then some level of randomness sufficient to ensure typicality in this regard. We describe and prove a large number of results in a new programme of research which aims to establish the (order theoretically) definable properties of the typical Turing degree, and the level of randomness required in order to guarantee typicality. A similar analysis can be made in terms of Baire category, where a standard form of genericity now plays the role that randomness plays in the context of measure. This case has been fairly extensively examined in the previous literature. We analyse how our new results for the measure-theoretic case contrast with existing results for Baire category, and also provide some new results for the category-theoretic analysis.

Barmpalias, George,Day, Adam R.,Lewis-Pye, Andy E. M.. The typical Turing degree[J]. PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY,2014-01-01,109:1-39.